Process for generating bidirectional reflectance distribution functions of gonioapparent materials with limited measurement data

ABSTRACT

A computer-implemented process for generating a bidirectional reflectance distribution function (BRDF) of a gonioapparent material containing effect flake pigments in a solid medium using limited measurement data, comprising the following steps: (A) acquiring and inputting into a computing device (1) photometric data and (2) the refractive index of the solid medium of the gonioapparent material; (B) converting any non-linear photometric data from step A) above to linear photometric data; (C) using the illumination angle and the reflective scattering angle associated with the linear photometric data and the refractive index of the medium to calculate corresponding effect flake angles; (D) fitting the linear photometric data and the effect angle data with an equation; (E) calculating the corresponding effect flake angle needed to calculate the BRDF being generated in step (F); and (F) generating the BRDF from the corresponding effect flake angle from step (E) and the equation developed in step (D).

FIELD OF THE INVENTION

The present invention relates to a process for the generation of thebidirectional reflectance distribution function (BRDF) of gonioapparentmaterials or surfaces, specifically those containing effect flakepigments, such as, metallic flake pigments or special effect flakepigments that typically are hue shifting interference pigments, withlimited measurement data.

BACKGROUND OF THE INVENTION

One the most general and well accepted means of describing the spectraland spatial reflective scattering properties of a material is by use ofthe bidirectional reflectance distribution function (BRDF). The BRDF isa fundamental description of the appearance of the surface of amaterial, and many other appearance attributes (such as, gloss, haze,and color) can be represented in terms of integrals of the BRDF overspecific geometries and spectral conditions. Specification of the BRDFis critical to the marketability of consumer products, such as,automobiles, cosmetics and electronics. The microstructure associatedwith a material affects the BRDF and specific properties can often beinferred from measurement of the BRDF. The angular distribution ofreflectively scattered light described by the BRDF can be used to renderthe appearance of materials or to predict the color appearance undervarying geometrical conditions. The quality of the rendering or colorprediction depends heavily on the accuracy of the BRDF of the materialsbeing rendered.

Gonioapparent objects or materials exhibit the characteristic ofchanging their appearance with change in illumination angle or viewingangle. Automotive finishes (paints) containing metallic flake pigmentsor special effect flake pigments, such as, pearlescent flake pigmentsare examples of gonioapparent materials. Unlike solid colors which canbe characterized at a single measurement geometry, gonioapparent colorsrequire measurements under a variety of illumination and viewinggeometries to describe their color appearance characteristics. Finishescontaining metallic flakes are generally characterized by making threecolor measurements at different aspecular angles. ASTM standard E-2194,which is hereby incorporated by reference, describes a standard practicefor multi-angle color measurement of metal flake pigmented materials.Finishes containing special effect flake pigments that are hue shiftingmaterials, such as, pearlescent pigments, also must be measured atmultiple geometries which vary in both aspecular angle and illuminationangle to characterize their color behavior.

In order to render objects on a video screen, or print media, orotherwise predict the color appearance of an object at a givenillumination and viewing geometry, the object's color at many thousandsof combinations of illumination and viewing angles must be calculated.

There are three basic techniques that have been used for the task ofcalculating all the required combinations of illumination and view.

-   -   1) The first technique is to actually measure the color of the        object at several thousand combinations of illumination and view        with an instrument such as a goniospectrophotometer, or        goniocolorimeter. This requires that a sufficient number of        measurements be made so that interpolation of the data to        predict the color of the object at intermediate geometries can        be done with sufficient accuracy. However, instruments with the        required geometric flexibility and photometric accuracy are        costly and very slow. Complete characterization of a single        color requires several hours of measurement time using this        technique.    -   2) A second technique is to develop a physical model of the        finish (color) and then use a technique, such as, radiative        transfer theory to calculate the color at all of the required        angular combinations. While techniques of this type can be used        to produce visually pleasing renderings, development and tuning        of the model to match the behavior of a physical standard is        extremely difficult and time consuming and may in fact be        impossible to do with sufficient fidelity.    -   3) The third technique is a combination of the first two with        the advantage of requiring far fewer measurements than the first        technique and a far less rigorous model of the finish than the        second technique. This third technique involves making a limited        number (typically 3-5) of color measurements of the object to be        rendered and then modeling the interpolation of this measured        data to the required angular combinations. This technique can        utilize 3-angle measurement data already contained in databases        typically used to store color characteristics of gonioapparent        materials. The models used to extrapolate this data to other        angular combinations do not require individual tuning and are        based on simple physical parameters of the surface of the        material.

For rendering or color prediction applications requiring measurement ofa vast array of colors, which match actual physical standards and arenot just “realistic looking” synthetic colors, the combination techniqueas described above is the preferred solution.

Alman (U.S. Pat. No. 4,479,718) led to the eventual wide spread adoptionof a three aspecular angle measurement system for characterization offinishes containing metal flake pigments in combination with absorbingand or scattering pigments. This measurement system serves as the basisfor such international standards as ASTM E-2194 and DIN 6175-2. Inpractice, this characterization approach also works well for formulationand control of finishes containing hue shifting (pearlescent) pigmentsonce pigmentation has been established.

While the concept of describing the gonioapparent color behavior of amaterial by measurements made at three aspecular angles is useful forformulation and control, and can be used to predict if a pair of sampleswill match under various measurement or viewing geometries, it is notwell suited to predict the absolute color of a material as themeasurement and viewing geometries change. For instance, while the samegeneral color change predicted by aspecular measurements hold as theillumination angle is changed, the magnitude of the color change is notwell predicted. FIG. 2 shows a plot of tristimulus value Y as a functionaspecular angle for a variety of illumination angles for an automotivepaint specimen containing metal flake pigment. While there is a trend toincreasing value of Y as the aspecular angle decreases, there are largedifferences in the absolute value of Y at a given aspecular angle as theillumination angle is changed.

A method is needed to predict the absolute color of a specimen, underany measurement or viewing geometry, from a limited (<10) set of colormeasurements.

SUMMARY OF THE INVENTION

The invention is directed to a computer-implemented process forgenerating a bidirectional reflectance distribution function (BRDF) orvariously normalized variants thereof of a gonioapparent materialcontaining effect flake pigments in a solid medium using limitedmeasurement data, comprising the following steps in any appropriateorder:

-   -   (A) acquiring and inputting into a computing device (1)        photometric data comprising spectral or calorimetric data of the        gonioapparent material being a function of an illumination angle        and a reflective scattering angle, wherein the data is obtained        by (a) measurements of the gonioapparent material, (b)        previously measured data of the gonioapparent material from a        data base containing measurements of the gonioapparent material        or (c) simulated data for a gonioapparent material and (2) the        refractive index of the solid medium of the gonioapparent        material;    -   (B) converting any non-linear photometric data from step (A)        above to a linear photometric data (e.g., converting non-linear        colorimetric L*, a*, b* data to linear X, Y, Z data);    -   (C) using the illumination angle and the reflective scattering        angle associated with the linear photometric data and the        refractive index of the medium to calculate corresponding effect        flake angles;    -   (D) fitting the linear photometric data and the effect flake        angles with an equation describing the linear photometric data        as a continuous function of effect flake angle via computer        implementation;    -   (E) calculating the corresponding effect flake angle from the        illumination angle, reflective scattering angle and refractive        index of the solid medium for each combination of illumination        and reflective scattering angle needed to calculate the BRDF        being generated in step (F); and    -   (F) generating the BRDF for each combination of illumination and        reflective scattering angle by calculating each value of the        BRDF from the corresponding effect flake angle from step (E)        above and the equation developed in step (D) above.    -   Also, the invention is directed to a system for generating BRDF        of the gonioapparent material wherein the system comprises a        computing device utilizing a computer readable program which        causes an operator to perform the above steps (A) through (F).

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and forfurther understanding of the advantages thereof, reference is now madeto the following detailed description taken in conjunction with thefollowing accompanying drawings:

FIG. 1 shows the geometry used in the determination of the BDRF.

FIG. 2 shows a plot of tristimulus value Y as a function aspecular anglefor a number of illumination angles for an automotive paint specimencontaining metal flake pigment.

FIG. 3 shows a schematic of how a ray of light is specularly reflectedfrom a metal flake suspended in a paint film.

FIG. 4 shows how, through the use of the algorithms embodied in thepresent invention, the illumination angle dependence of the data shownin FIG. 2 can be eliminated.

FIG. 5 shows the three measurements used in step (B) of the exampledemonstration of the present invention.

FIG. 6 shows the conversion of the data in FIG. 5 from being a functionof aspecular angle to being a function of effect flake angle.

FIG. 7 shows the fit of the example effect flake angle data with anequation of the form:

$\begin{matrix}{\rho_{\theta_{f}}^{\prime} = {{A \times \exp^{({- \frac{\theta_{f}}{B}})}} + C}} & {{Equation}\mspace{14mu}(1)}\end{matrix}$

FIG. 8 shows how the curve fit compares with the rest of the measureddata that was not used in the calculation of the curve fit.

FIG. 9 shows prediction of the measured data from back calculation ofthe curve fit.

FIG. 10 shows a comparison of the measured and fit data.

FIG. 11 shows a comparison of the measured and fit data, along with alinear regression fit to the data.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The features and advantages of the present invention will be morereadily understood, by those of ordinary skill in the art, from readingthe following detailed description. It is to be appreciated that certainfeatures of the invention, which are, for clarity, described above andbelow in the context of separate embodiments, may also be provided incombination in a single embodiment. Conversely, various features of theinvention that are, for brevity, described in the context of a singleembodiment, may also be provided separately or in any sub-combination.In addition, references in the singular may also include the plural (forexample, “a” and “an” may refer to one, or one or more) unless thecontext specifically states otherwise.

The use of numerical values in the various ranges specified in thisapplication, unless expressly indicated otherwise, are stated asapproximations as though the minimum and maximum values within thestated ranges were both preceded by the word “about.” In this manner,slight variations above and below the stated ranges can be used toachieve substantially the same results as values within the ranges.Also, the disclosure of these ranges is intended as a continuous rangeincluding every value between the minimum and maximum values.

All patents, patent applications and publications referred to herein areincorporated by reference in their entirety.

The following terms are used herein:

“Effect flake pigments” include metallic flake pigments, such asaluminum flake, coated aluminum flake, gold flake, copper flake, and thelike and also includes special effect flake pigments that cause a hueshift, such as, pearlescent pigments, such as, coated mica flake, coatedAl₂O₃ flake, coated glass flake, coated SiO₂ flake, and the like.

“Aspecular angle” the viewing angle measured from the speculardirection, in the illuminator plane unless otherwise specified. Positivevalues of aspecular angle are in the direction toward the illuminatoraxis.

“Effect flake angle” the angle between the surface normal of the effectflake pigment and the surface normal of the sample specimen.

“Gonioapparent”—pertaining to change in appearance with change inillumination or viewing angle.

“Gonioappearance”—the phenomenon in which the appearance of a specimenchanges with change in illumination or viewing angle.

ASTM Standard E 2387-05, which is hereby incorporated by reference,describes procedures for determining the amount and angular distributionof reflective optical scatter from a surface, and provides precisedefinitions of many of the terms used in the description of the presentinvention. Definitions for terms not found in ASTM Standard E 2387-05will be found in ASTM terminology Standard E 284, which is herebyincorporated by reference.

BRDF (bidirectional reflectance distribution function) means acollection of photometric data of any material (herein meaninggonioapparent material) that will describe photometric reflective lightscattering characteristics of the material as a function of illuminationangle and reflective scattering angle. It is one of the most general andwell accepted means of describing the spectral and spatial reflectivescattering properties of a gonioapparent material and provides afundamental description of the appearance of a material and many otherappearance attributes (such as, gloss, haze, and color) can berepresented in terms of integrals of the BRDF over specific geometriesand spectral conditions.

The BRDF is dependent on wavelength, incident direction, scatterdirection and polarization states of the incident and reflectivelyscattered fluxes. The BRDF is equivalent to the fraction of incidentflux reflectively scattered per unit of projected angle;

$\begin{matrix}{{f_{r}\left( {\Theta_{i}\Theta_{s}\lambda} \right)} = \frac{\mathbb{d}{L_{(s)}\left( {\Theta_{i}\Theta_{s}\lambda} \right)}}{\mathbb{d}{E_{(i)}\left( {\Theta_{i}\lambda} \right)}}} & {{Equation}\mspace{14mu}(2)}\end{matrix}$

Where the subscripts i and r denote incident and reflected respectively,Θ=(θ,φ) is the direction of light propagation, λ is the wavelength oflight, L is radiance, and E is irradiance. The geometry used by the BRDFis shown in FIG. 1. Where Θ_(i) and θ_(r) are the illumination andreflective scattering vectors respectively. θ_(p) is the specimensurface normal vector. θ_(i) and θ_(r) are the illumination andreflective scattering polar angles respectively, and φ_(i) and φ_(r) arethe illumination and reflective scattering azimuthal anglesrespectively. “x, y, z” are Cartesian coordinate axes.

In practice, the BRDF of a gonioapparent material is often expressed asdirectional reflectance factor R_(d) which is the ratio of the specimenBRDF to that for a perfect reflecting diffuser (defined as 1/π), givenby:R_(d)=πf,  Equation (3)

For color work, the BRDF is often expressed as the colorimetric BRDF.The colorimetric BRDF consists of three color coordinates as a functionof the scattering geometry. Using CIE color matching functions [ x(λ),y(λ), z(λ)] for one of the CIE standard colorimetric standard observersand a CIE standard illuminant S(λ), the colorimetric BRDF is defined as:f _(r(color,X)) =k∫ _(λ) f _(r)(λ)S(λ) xd(λ)  Equation (4)f _(r(color,Y)) =k∫ _(λ) f _(r)(λ)S(λ) yd(λ)  Equation (5)f _(r(color,Z)) =k∫ _(λ) f _(r)(λ)S(λ) zd(λ)  Equation (6)

The normalizing factor k is defined as:k=(∫_(λ) S(λ) y (λ)dλ)¹  Equation (7)

The specific illuminant (for example, CIE Standard Illuminant D65), setof color matching functions (for example, CIE 1964 Standard Observer)and the color system (for example, CIELAB) must be specified andincluded with any data.

For purposes of this patent, the term BRDF, as designated by the symbolf_(r) is meant to include the formal definition of BRDF in terms ofreflectance as outlined by, F. E. Nicodemus, J. C. Richmond, J. J. Hsia,I. W. Ginsberg, and T. Limperis, “Geometrical considerations andnomenclature for reflectance,” NBS Monograph 160 (National Bureau ofStandards, Washington, D.C., 1977) as well as any of the normalizedvariants of the BRDF based on reflectance factor. These variantsinclude, but are not limited to, directional reflectance factor f_(r(R)₂ _(,λ)), or any of the colorimetric BRDF variants, linear (e.g.f_(r(color,X)), f_(r(color,Y)), f_(r(color,Z))) or non-linear (e.g.f_(r(color,L)), f_(r(color,M)), f_(r(color,N))), including BRDFsgenerated in RGB space (e.g. f_(r(RGB,R)), f_(r(RGB,G)), f_(r(RGB,B)))often used in the graphic rendering applications. The basic steps of theprocess of the present invention are the same for all of the BRDFvariants with the differences existing in pre-, or post-processing ofthe data, to convert from one color space to another, known to oneskilled in the art.

Changes in the intensity of reflectively scattered light from a materialcontaining effect flake pigments with change in illumination angle orviewing angle are due to the angular distribution of the effect flakepigments in the finish. Typically in a painting process, during dryingof paint film applied to a substrate, the effect flake pigmentssuspended in the paint binder tend to orient themselves roughly parallelto the surface of the substrate being coated. These pigments act as tinymirrors to specularly reflect light that strikes the pigments. Thehigher the percentage of effect flake pigment flakes that are orientedso as to act as specular reflectors for a given geometry, the higher thereflected intensity. Other means of specimen preparation of materialscontaining effect flake pigments will also orient the effect flakepigment flakes to some extent, depending on the method of specimenpreparation, e.g., injection molding, or casting.

The primary reason for the differences in reflected intensity at a givenaspecular angle as a function of illumination angle, as shown in FIG.2., is that the aspecular geometry is calculated with respect to thesurface normal of the material. However due to refraction effects in thepaint film, this does not directly describe the angular distribution ofthe effect flake pigments in the paint film. If the effect flakepigments were suspended in a medium with a refractive index of (1.0)then angle of illumination would not matter; however, this is not thecase, and a means to correct for film refraction effects is needed.

Through the use of a process to correct for refraction of the light asit enters and exits the specimen surface, the illumination angledependence of data in FIG. 2 can be eliminated as shown in FIG. 4.

The following is a step-by-step description of the process used toremove the illumination angle dependence of the data apparent in FIG. 2,and subsequent generation of the BRDF from the processed data. In thisdescription, it is assumed that the specimen plane lies in the x-y planeof a Cartesian coordinate system.

In the first step (A) of the process to generate the BRDF of thespecimen, photometric data ρ′(Θ_(i)Θ_(r)), comprising either spectral orcolorimetric data as a function of the illumination angle Θ_(i) andreflective scattering angle Θ_(r), are acquired and input into acomputing device. This data may be acquired from a database wherein thisphotometric data has been previously measured, typically at threedifferent angles, or by actual measurements taken of the specimen,typically at three different angles, or as simulated data for aspecimen, typically at three different angles. The combinations ofillumination angle and reflective scattering angle typically used are45:−30(as 15), 45:0(as 45) and 45:65(as 110) resulting in aspecularangles of 15, 45 and 110 degrees. Other appropriate combinations ofillumination angle and reflective scattering angle resulting in similaraspecular angle combinations may also be used, such as those resultingin aspecular angle combinations of 15, 45, and 75 degrees and 25, 45 and75 degrees.

The photometric measurements can be acquired by instruments, such as,the Model GCMS Goniospectrophotometric Measurement System available fromMurakami Color Research Laboratory, Tokyo, Japan, or the Model MultiFX10spectrophotometer available from Datacolor International Incorporated,Lawrenceville, N.J. or the Model MA68 spectrophotometer available fromX-Rite Incorporated, Grandville, Mich.

The refractive index of the matrix containing the effect flake pigments,such as a paint, is determined either by measurement with a device, suchas, the Model 2010 Prism Coupler available from Metricon Corp.,Pennington N.J. or by retrieval from a database.

In step (B) of the process, any non-linear photometric data from step(A) of the process must be converted to a linear photometric basis. Forexample, any photometric data expressed as CIELAB color coordinates,also commonly referred to as L*a*b* or Lab, must be converted to linearX, Y, Z tristimulus space. L*, a* b* color values are well known tothose skilled in the art and represent coordinates in visual uniformcolor space and are related to X, Y and Z tristimulus values by thefollowing equations which have been specified by the InternationalCommittee on Illumination:

L* defines the lightness axis

$\begin{matrix}{L^{*} = {116\left\lbrack {{f\left( \frac{Y}{Y_{0}} \right)} - \frac{16}{116}} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 8} \right)\end{matrix}$

a* defines the red green axis

$\begin{matrix}{a^{*} = {500\left\lbrack {{f\left( \frac{X}{X_{0}} \right)} - {f\left( \frac{Y}{Y_{0}} \right)}} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 9} \right)\end{matrix}$

b* defines the yellow blue axis

$\begin{matrix}{b^{*} = {200\left\lbrack {{f\left( \frac{Y}{Y_{0}} \right)} - {f\left( \frac{Z}{Z_{0}} \right)}} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

where

-   -   X_(o), Y_(o) and Z_(o) are the tristimulus values of the perfect        white for a given illuminant.

and where

-   -   f(Y/Y₀)=(Y/Y₀)^(1/3) for Y/Y₀ greater than 0.008856 and    -   f(Y/Y₀)=7.787(Y/Y₀)+16/116 for Y/Y₀ less than or equal to        0.008856; f(X/X₀) and f(Z/Z₀) are similarly defined.        In step (B) of the process using the above equations, the L* a*        b* values for each of the angle combinations utilized are        converted into tristimulus X, Y, and Z values as outlined in the        following section of computer pseudo-code.

If L* < 7.99962    Then YYN = L*/903.3 Equation (11)    Else YYN =((L* + 16)/116)³ Equation (12) End If    Y = YYN × Y_(o) Equation (13)   If YYN > 0.008856       Then FYYN = YYN^(1/3) Equation (14)      Else FYYN = 7.787 × YYN + 0.13793 Equation (15)    End If    FXXN= a* / 500 +FYYN Equation (16)    If FXXN > 0.206893       Then XXN =FXXN³ Equation (17)       Else XXN = (FXXN − 0.13793) /7.787 Equation(18)    End If    X = XXN * X₀ Equation (19)    FZZN =FYYN − b*/200Equation (20)    If FZZN > 0.206893       Then ZZN = FZZN³ Equation (21)      Else ZZN = (FZZN − 0.13793) /7.787 Equation (22)    End IF    Z =ZZN * Z₀ Equation (23) where    X_(o), Y_(o) and Z_(o) are the abovedescribed tristimulus values. and where    YYN, FYYN, XXN, FXXN, ZZN,FZZN are intermediate    variables used only during the calculationThe above equations are shown in ASTM Standard E 308, which is herebyincorporated by reference.

In step (C) of the process the illumination angle dependence of the datais eliminated by converting the data from an aspecular angle basis to aneffect flake angle basis. To accomplish this, calculate the unitdirection vectors i, r and p for the illumination ray,Θ_(i)=(θ_(i),φ_(i)), reflectively scattered ray, Θ_(r)=(θ_(r),φ_(r)),and specimen normal Θ_(p) respectively.i=(α_(i),β_(i),γ_(i))  (Equation 24)r=(α_(r),β_(r),γ_(r))  (Equation 25)p=(α_(p),β_(p),γ_(p))  (Equation 26)where α²+β²+γ²=1  (Equation 27)for each of geometries represented in the data acquired in step (A).

The direction cosines α, β, γ are determined from the polar andazimuthal angles θ and φ as follows:α=sin θ cos φ  Equation (28)β=sin θ sin φ  Equation (29)γ=cos θ  Equation (30)

Next, calculate the cosine of the included angle τ_(ip) between theincident ray direction vector i and the specimen surface normal p.cos τ_(ip)=α_(i)*α_(p)+β_(i)*β_(p)+γ_(i)*γ_(p)  (Equation 31)

Similarly, also calculate the cosine of the included angle τ_(rp)between the reflectively scattered ray direction vector r and thespecimen surface normal direction vector p,cos τ_(rp)=α_(r)*α_(p)+β_(r)*β_(p)+γ_(r)*γ_(p)  (Equation 32)The angles τ_(ip) and τ_(rp) are shown in two dimensions in FIG. 3 asangles ∠ABD and ∠CB′D′ respectively.

As illustrated in two dimensions in FIG. 3, as the incident ray ABpasses through the surface of the specimen matrix, it is refractedtoward the surface normal BF, as ray BG. Similarly the reflectivelyscattered ray GB′ is refracted away from the surface normal B′D′ as itpasses out of the specimen matrix as ray B′C. Rays BG and GB′ representthe in-matrix illumination ray Θ_(i′) and the in-matrix reflectivelyscattered ray Θ_(r′) respectively. Next determine the direction vectori′ of in-matrix illumination ray Θ_(i′) and the direction vector s′ ofin-matrix reflective scattered ray Θ_(r′) from angles ∠ABD, τ_(ip)above, and ∠CB′D′, τ_(rp) above in conjunction with the refractive indexof the specimen matrix η₂ and the refractive index of air η₁ using thevector form of Snell's law:

$\begin{matrix}{{{\cos\;\tau_{i^{\prime}p}} = \sqrt{1 - {\left( \frac{\eta_{1}}{\eta_{2}} \right)^{2}\left( {1 - \left( {\cos\;\tau_{ip}} \right)^{2}} \right)}}}{i_{\alpha,\beta,\gamma}^{\prime} = {{\left( \frac{\eta_{1}}{\eta_{2}} \right) i_{\alpha,\beta,\gamma}} + {\left( \begin{matrix}{{\cos\;\tau_{i^{\prime}p}} -} \\{\frac{\eta_{1}}{\eta_{2}}\cos\;\tau_{ip}}\end{matrix} \right) p_{\alpha,\beta,\gamma}}}}} & \left( {{{{Equations}\mspace{14mu} 33}\&}\mspace{11mu} 34} \right)\end{matrix}$and

$\begin{matrix}{{{\cos\;\tau_{r^{\prime}p}} = \sqrt{1 - {\left( \frac{\eta_{1}}{\eta_{2}} \right)^{2}\left( {1 - \left( {\cos\;\tau_{rp}} \right)^{2}} \right)}}}{r_{\alpha,\beta,\gamma}^{\prime} = {{\left( \frac{\eta_{1}}{\eta_{2}} \right)r_{\alpha,\beta,\gamma}} + {\begin{pmatrix}{{\cos\;\tau_{r^{\prime}p}} -} \\{\frac{\eta_{1}}{\eta_{2}}\cos\;\tau_{rp}}\end{pmatrix}p_{\alpha,\beta,\gamma}}}}} & \left( {{{{Equations}\mspace{14mu} 35}\&}\mspace{11mu} 36} \right)\end{matrix}$

For a effect flake pigment flake to act as a specular reflector for aparticular geometry, its surface normal must bisect the angle formed bythe in-matrix illumination ray and the in-matrix reflective scatteringray. The surface normal vector of the effect flake pigment flake iscalculated as follows:Θ_(f) =i′ _(α,β,γ) +r′ _(α,β,γ)  (Equation 37)converting Θ_(f) from its Cartesian form Θ_(f)=(α_(f),β_(f),γ_(f)) toits spherical coordinate form Θ_(f)=(θ_(f), φ_(f)), the polar angleθ_(f) describes the angle of the effect flake pigment flake surfacenormal to the specimen surface normal which we call the effect flakeangle.

In step (D) of the process, the linear photometric data, ρ′, from step(B) and the effect flake angle data, θ_(f), from step (C) are fit viacomputer implementation with an equation which describes the photometricdata as a continuous function of the effect flake angle. A usefulfunctional form for this equation is the exponential decay plus constantequation of the form:

$\begin{matrix}{\rho_{\theta_{f}}^{\prime} = {{A \times \exp^{({- \frac{\theta_{f}}{B}})}} + C}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$Where A, B, C are weighting constants calculated in the curve fittingprocess.

While the exponential decay function is well suited to fit typical data,there are equations of other functional forms which may also be useful.

In step (E) of the process, for each geometry to be calculated in theBRDF, the effect flake angle, θ_(f), for the geometry is calculatedusing the same procedure as described in step (C) of the processsubstituting the BRDF geometry for the measurement geometry.

In step (F) of the process, for each geometry to be calculated in theBRDF, the value of the BRDF is calculated by solving the equation:

$\begin{matrix}{f_{r{({\theta_{i},\phi_{i},\theta_{r},\phi_{r}})}} = {{A \times \exp^{({- \frac{\theta_{f{({\theta_{i},\phi_{i},\theta_{r},\phi_{r},\eta_{2}})}}}{B}})}} + C}} & \left( {{Equation}\mspace{14mu} 38} \right)\end{matrix}$using the value of θ_(f(θ) _(i) _(,φ) _(i) _(,θ) _(r) _(,φ) _(r) _(,η) ₂₎ from step (E) of the process and weighting coefficients A, B, C fromstep (D) of the process.

In an optional step (G) the linear BRDF values calculated in steps (A-F)can be converted to non-linear BRDF values as necessary, e.g. if thedesired final BRDFs are in non-linear L*a*b* space they must becalculated from linear BRDFs in XYZ space. The mathematics of thisconversion are known to those skilled in the art.

Once the BRDF has been generated, the data contained in the BRDF can beused for a wide variety of purposes. One of the most widely used uses ofBRDF data is for rendering the appearance of an object on some displaymedia such as a video display device, print media, photographic mediaand the like. The processes and computational algorithms for generatingdisplay R,G,B values based on BRDF, integrating the display R,G,B valueswith shapes of the object, and rendering appearance of the object arewell known to those skilled in the art. There are also a variety ofcommercial and proprietary computer programs available to do the objectrendering including U-Drive from Bunkspeed of Los Angeles, Calif., OpusRealizer from Opticore AB of Gothenburg, Sweden, Maya from Autodesk ofSan Rafael, Calif., and the like.

An example of a process for rendering appearance of an object based onBRDF can be described briefly below. A user first selects color to berendered for the object. Color data corresponding to the selected coloris retrieved from a color database or acquired by measuring the objectusing methods well known to those skilled in the art. The process isdescribed in detail by Rupieper et al. (U.S. Pat. No. 6,618,050) andVoye et al. (U.S. Pat. No. 6,977,650). Line 62 of column 4 through line44 of column 8 of the aforementioned U.S. Pat. No. 6,618,050, and line 5of column 6 through line 45 of column 11 of the aforementioned U.S. Pat.No. 6,977,650 are incorporated herein by reference. In brief, images orsurface topographies of the object are facetted into a sufficient numberof polygons to generate polygon data. One or more illumination andviewing angles can be selected or simulated to generate illumination andviewing angle data. The polygon data, the illumination and viewing angledata, and BRDF data based on this invention are integrated into acomputing process to convert X,Y,Z BRDF color data of the selected colorinto display X,Y,Z data that reflect appearance of the selected colorunder said illumination and viewing angles. The display X,Y,Z data canbe scaled up or down as determined necessary by those skilled in theart. Display R,G,B data can be defined based on the display X,Y,Z data.Depending on the display devices, the display R,G,B data may need to becalibrated or adjusted based on the display device profile. The processis repeated for each of plurality of pixels of the polygon and each ofthe illumination and viewing angles, and each of the sufficient numberof polygons. The appearance of the object is then displayed via thedisplay device.

In addition to its use in rendering applications, the data contained inthe BDRF generated by the present invention can be used for a variety ofother uses. The absolute color or reflectance data can be used inconjunction with pigment mixture models to aid in the formulation ofpaint finishes or molded plastic products containing effect flakepigments, to assess and insure color match at a wide variety ofillumination and viewing conditions. BDRF data can be used to predictthe visual differences between surfaces, coated with the same ordifferent materials, presented to the observer at slightly differentgeometries, e.g. the BRDF data can be used to assess color match betweenan automobile body and body fascia such as bumper covers or other trim.Color difference data calculated from the BRDF data of two or morespecimens can be used for a variety of different color shading andcontrol applications. While the example uses of BRDF data cited aboverepresent typical uses of this data it is not meant to be a restrictiveor complete list of uses for the BRDF data.

EXAMPLE

The present invention is further defined in the following Example. Itshould be understood that this Example, while indicating preferredembodiments of the invention, is given by way of illustration only. Fromthe above discussion and this Example, one skilled in the art canascertain the essential characteristics of this invention, and withoutdeparting from the spirit and scope thereof, can make various changesand modifications of the invention to adapt it to various uses andconditions.

Example 1

The following example demonstrates and illustrates the steps required togenerate an L* colorimetric BRDF for an automotive paint samplecontaining metal flake effect flake pigments. The L* colorimetric axisis used as an example only, and the same basic steps are required forcalculation of other colorimetric BRDF axes or directional reflectancefactor.

In step (A) of the process, the sample specimen is placed in a ModelGCMS Goniospectrophotometric Measurement System which has beencalibrated according to the manufacture's established procedure. L*a*b*measurements of the specimen are made at the following set ofillumination and viewing geometries:

1) θ_(i1)=45 degrees, φ_(i1)=0 degrees, θ_(s1)=30 degrees, φ_(s1)=180degrees

2) θ_(i2)=45 degrees, φ_(i2)=0 degrees, θ_(s2)=0 degrees, φ_(s2)=0degrees

3) θ_(i3)=45 degrees, φ_(i3)=0 degrees, θ_(s3)=75 degrees, φ_(s3)=0degrees

These geometries represent measurements at aspecular angles of 15degrees, 45 degrees, and 110 degrees respectively. The L*a*b*measurements acquired are as follows:

1) L*₁=33.90, a*₁=−7.58, b*₁=−36.61

2) L*₂=12.29, a*₂=1.19 b*₂=−25.77

3) L*₃=3.07, a*₃=2.09 b*₃=−13.24

Additionally the refractive index of the paint matrix is measured on aMetricon Model 2010 Prism Coupler which has been set-up and calibratedaccording to the manufacture's established recommended procedure. Therefractive index of the specimen matrix η₂ is measured to be 1.5109.

In addition to the data above, L*a*b* measurements were also made at avariety of other measurement geometries which although not required forthe present invention will be used later in this example to demonstratethe utility of the present invention.

In step (B) of the process, look at the photometric data acquired instep (A) and find that the data is non-linear L*a*b* data and thereforemust be converted to a linear basis, in this case conversion totristimulus X,Y,Z space is appropriate. Using equations (11-23) theL*,a*,b* data is converted to X,Y,Z data. Since this example onlydemonstrates generation of an L* colorimetric BRDF only the Y values arecalculated as the L* value has no X, or Z component. The Y valuesassociated with the three acquired measurements above are as follows:

1) Y₁=7.96

2) Y₂=1.45

3) Y₃=0.34

FIG. 5 shows a plot of this Y data versus aspecular angle.

Using the equations outlined in step (C) of the process description, themeasurement geometries associated with the photometric data areconverted to an effect flake angle basis. The effect flake anglesassociated with each of the measurements are as follows:

1) θ_(f1)=4.25 degrees

2) θ_(f2)=13.82 degrees

3) θ_(f3)=32.07 degrees

FIG. 6 shows a plot of the conversion on the Y data from an aspecularangle basis (square symbols) to an effect flake angle basis (circularsymbols).

In step (D) of the process the photometric p data (Y data in thisexample) from step (B) of the process, and the effect flake angle data,θ_(f), from step (C) of the process are fit via computer implementationwith an equation of the form:

$\begin{matrix}{\rho_{\theta_{f}}^{\prime} = {{A \times \exp^{({- \frac{\theta_{f}}{B}})}} + C}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$The coefficients of the equation, A, B, C are as follows:A=17.86, B=1.45, C=0.34

FIG. 7 shows a graph of the function above plotted with the coefficientsabove as a function of effect flake angle. Note that the experimentaldata points (circular symbols) are fit exactly.

In step (E) of the process, for each geometry to be calculated in theBRDF, the effect flake angle for the geometry is calculated using thesame procedure as described in step (C) of the process substituting theBRDF geometry for the measurement geometry. For purposes of thisexample, use the measurement geometries associated with the extrameasurements made in step (A) of the example above as the BRDFgeometries to be modeled.

FIG. 8 shows a plot of the extra measurements acquired in step (A) aboveafter conversion to effect flake angle basis superimposed on top of thecurve fit through the original data. Note the excellent agreementbetween the predicted data (solid curve) and the measured data (diamondshaped symbols). Since the measured data represents measurements takenat a wide variety of illumination angles, this demonstrates that theprocess outlined in this invention is capable of removing theillumination angle dependence of the data as shown in FIG. 2). FIG. 9)shows the BRFD predicted data plotted once again as a function ofaspecular angle. FIG. 10) shows a comparison of the measured and fitdata plotted versus aspecular angle. FIG. 11) shows a plot of measuredversus predicted data along with linear regression fit of the data. Theslope of the regression fit being close to a value of 1.0, the value ofthe intercept being close to a value of 0.0, along with thegoodness-of-fit statistic R² value being close to 1.0 indicate that themodel and hence, the procedure as outlined here is useful predicting theBRDF of a specimen containing effect pigments with a very limited (inthis case 3 measurement points) amount of data. As is to be expected,the largest deviations in predicted data vs. measured data occur wherethe slope of the fitted curve is steep, the small deviations shown hereare not expected to be visually objectionable in a rendering producedusing this technique.

This same procedure has also be shown to work well with finishescontaining most hue shifting pigments, such as coated mica flakes, andthe like.

1. A computer-implemented process for generating a bidirectionalreflectance distribution function (BRDF) of a gonioapparent materialspecimen containing effect flake pigments in a solid medium usinglimited measurement data, comprising the following steps in anyappropriate order: (A) acquiring and inputting into a computing device(1) photometric data comprising spectral or colorimetric data of thegonioapparent material being a function of an illumination angle and areflective scattering angle, wherein said data being obtained by (a)measurements of the gonioapparent material, (b) previously measured dataof the gonioapparent material from a data base containing measurementsof the gonioapparent material or (c) simulated data for a gonioapparentmaterial and (2) the refractive index of the solid medium of thegonioapparent material; (B) converting any non-linear photometric datafrom step (A) above to linear photometric data; (C) using theillumination angle and the reflective scattering angle associated withthe linear photometric data and the refractive index of the medium tocalculate corresponding effect flake angles; (D) fitting the linearphotometric data and the effect flake angle data with an equationdescribing the linear photometric data as a continuous function ofeffect flake angle via computer implementation; (E) calculating thecorresponding effect flake angle from the illumination angle, reflectivescattering angle and refractive index of the solid medium for eachcombination of illumination and reflective scattering angle needed tocalculate the BRDF being generated in step (F); and (F) generating theBRDF for each combination of illumination and reflective scatteringangle by calculating each value of the BRDF from the correspondingeffect flake angle from step (E) above and the equation developed instep (D) above.
 2. The computer-implemented process of claim 1 whereinthe BRDF generated is used to render the appearance of the specimen. 3.The computer-implemented process of claim 1 wherein normalized variantsof BRDF of gonioapparent materials are determined.
 4. Thecomputer-implemented process of claim 1 wherein the BRDF generated isused to calculate the absolute colorimetric or spectral reflectance datafor the specimen at angles not specifically measured in step (A).
 5. Thecomputer-implemented process of claim 1 wherein the BRDF generated isused to calculate color difference, or spectral reflectance differencedata for a pair of specimens at geometries not specifically measured instep (A).
 6. The computer-implemented process of claim 1 wherein threecombinations of illumination and reflective scattering angle are used instep (A) and subsequent steps.
 7. The computer-implemented process ofclaim 6 wherein up to ten combinations of illumination and reflectivescattering angle are used in step (A) and subsequent steps.
 8. Thecomputer-implemented process of claim 6 wherein the three combinationsof illumination and reflective scattering angle result in detection ataspecular angles of 15, 45 and 110 degrees.
 9. The computer-implementedprocess of claim 2 where the BRDF are combined with spatial textureinformation to render the appearance of the specimen.
 10. Thecomputer-implemented process of claim 2 where the specimen is renderedon a video display device.
 11. The computer-implemented process of claim2 where the specimen is rendered on print media.
 12. Thecomputer-implemented process of claim 2 where the specimen is renderedon photographic media.
 13. The computer-implemented process of claim 1wherein the pigments in the gonioapparent material are aluminum flakepigments.
 14. The computer-implemented process of claim 1 wherein thepigments in the gonioapparent material are hue shifting flake pigments.15. The computer implemented process of claim 1 that includes anoptional step (G) that converts the BRDF to a non-linear basis.
 16. Asystem for generating a bidirectional reflectance distribution function(BRDF) of a gonioapparent material containing effect flake pigments in asolid medium using limited measurement data, said system comprising (1)a computing device; (2) a computer readable program which causes anoperator and the computing device to perform the following: (A)acquiring and inputting into a computing device (1) photometric datacomprising spectral or colorimetric data of the gonioapparent materialbeing a function of an illumination angle and a reflective scatteringangle, wherein said data being obtained by (a) measurements of thegonioapparent material, (b) previously measured data of thegonioapparent material from a data base containing measurements of thegonioapparent material or (c) simulated data for a gonioapparentmaterial and (2) the refractive index of the solid medium of thegonioapparent material; (B) converting any non-linear photometric datafrom step A) above to linear photometric data; (C) using theillumination angle and the reflective scattering angle associated withthe linear photometric data and the refractive index of the medium tocalculate corresponding effect flake angles; (D) fitting the linearphotometric data and the effect flake angle data with an equationdescribing the linear photometric data as a continuous function ofeffect flake angle via computer implementation; (E) calculating thecorresponding effect flake angle from the illumination angle, reflectivescattering angle and refractive index of the solid medium for eachcombination of illumination and reflective scattering angle needed tocalculate the BRDF being generated in step (F); and (F) generating theBRDF for each combination of illumination and reflective scatteringangle by calculating each value of the BRDF from the correspondingeffect flake angle from step (E) above and the equation developed instep (D) above.
 17. The system of claim 16 that includes an optionalstep (G) that converts the BRDF to a non-linear basis.